1 Apartments in Vitoria

Making a new factor variable for elevators named elevatorF.

VIT2005 <- VIT2005 %>% 
  mutate(elevatorF = as.factor(elevator)) %>% 
  glimpse()
Rows: 218
Columns: 16
$ totalprice     <dbl> 228000.0, 409000.0, 200000.0, 180000.0, 443600.0, 1730…
$ area           <dbl> 75.31, 100.65, 88.87, 62.61, 146.15, 77.21, 77.04, 62.…
$ zone           <fct> Z45, Z31, Z52, Z62, Z31, Z11, Z47, Z48, Z11, Z11, Z37,…
$ category       <fct> 4B, 3B, 3A, 4A, 3A, 4B, 3A, 3B, 4B, 3B, 4B, 4B, 2B, 4B…
$ age            <int> 33, 5, 14, 41, 22, 35, 14, 36, 37, 11, 36, 10, 7, 9, 1…
$ floor          <int> 3, 7, 8, 3, 6, 4, 6, 3, 4, 5, 6, 3, 4, 2, 6, 4, 4, 4, …
$ rooms          <int> 5, 5, 5, 4, 7, 5, 4, 4, 4, 4, 6, 5, 6, 5, 5, 4, 4, 5, …
$ out            <fct> E100, E50, E50, E50, E100, E50, E50, E100, E25, E50, E…
$ conservation   <fct> 2B, 1A, 1A, 2A, 1A, 1A, 1A, 3A, 2A, 1A, 2B, 1A, 1A, 1A…
$ toilets        <int> 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, …
$ garage         <int> 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, …
$ elevator       <int> 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, …
$ streetcategory <fct> S3, S5, S2, S3, S4, S4, S3, S3, S4, S4, S2, S3, S4, S4…
$ heating        <fct> 3A, 4A, 3A, 1A, 4A, 3A, 4A, 3A, 3A, 3A, 4A, 3A, 3B, 3A…
$ storage        <int> 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, …
$ elevatorF      <fct> 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, …

1.1 Exercise

  • Create a scatterplot of totalprice versus area.
ggplot(data = VIT2005, aes(x = area, y = totalprice)) +
  geom_point() + 
  theme_bw() +
  labs(x = "Area in square meters", y = "Price in Euros")
No this is Boo playing a joke

Figure 1.1: No this is Boo playing a joke

To refer to the Figure 1.1, use the syntax \@ref(fig:label).

  • Compute \(r\) using the formula given in (1.1) and verify your answer using the built-in R function cor().
\[\begin{equation} r = \frac{1}{n-1}\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{s_x}\right)\cdot\left(\frac{y_i - \bar{y}}{s_y}\right) \tag{1.1} \end{equation}\] 
values <- VIT2005 %>% 
  mutate(x_xbar = area - mean(area), y_ybar = totalprice - mean(totalprice),
         zx = x_xbar/sd(area), zy  = y_ybar/sd(totalprice)) %>% 
  select(area, x_xbar, zx, totalprice, y_ybar, zy)
head(values)
    area      x_xbar           zx totalprice     y_ybar         zy
1  75.31 -13.3928463 -0.645972464     228000  -52741.52 -0.7610779
2 100.65  11.9471576  0.576243069     409000  128258.48  1.8508128
3  88.87   0.1671589  0.008062516     200000  -80741.52 -1.1651273
4  62.61 -26.0928433 -1.258526968     180000 -100741.52 -1.4537340
5 146.15  57.4471500  2.770828263     443600  162858.48  2.3501024
6  77.21 -11.4928448 -0.554330357     173000 -107741.52 -1.5547463
values %>% 
  summarize(r = sum(zx*zy)/(length(zx) - 1))
          r
1 0.8092125
#
#  Built In Function
VIT2005 %>% 
  summarize(r = cor(totalprice, area))
          r
1 0.8092125
  • Create a linear model object named mod1 by regressing totalprice on to area.
mod1 <- lm(totalprice ~ area, data = VIT2005)
mod1

Call:
lm(formula = totalprice ~ area, data = VIT2005)

Coefficients:
(Intercept)         area  
      40822         2705
  • Write the least squares regression equation.

\[\widehat{\text{totalprice}} = 4.0822416\times 10^{4} + 2704.7510279\cdot \text{area}\]

  • Use the coefficients from mod1 and mutate() to compute the \(\hat{y}\) and \(e_i\) values.
values <- VIT2005 %>%
  mutate(yhat = coef(mod1)[1] + coef(mod1)[2]*area,
         e = totalprice - yhat) %>% 
  select(totalprice, yhat, area, e)
head(values)
  totalprice     yhat   area          e
1     228000 244517.2  75.31 -16517.209
2     409000 313055.6 100.65  95944.389
3     200000 281193.6  88.87 -81193.647
4     180000 210166.9  62.61 -30166.879
5     443600 436121.8 146.15   7478.238
6     173000 249656.2  77.21 -76656.240
# Or
values <- VIT2005 %>% 
  mutate(yhat = predict(mod1), e = resid(mod1)) %>% 
  select(totalprice, yhat, area, e)
head(values)
  totalprice     yhat   area          e
1     228000 244517.2  75.31 -16517.209
2     409000 313055.6 100.65  95944.389
3     200000 281193.6  88.87 -81193.647
4     180000 210166.9  62.61 -30166.879
5     443600 436121.8 146.15   7478.238
6     173000 249656.2  77.21 -76656.240
  • Find the least squares coeffients for regressing totalprice on to area using the formulas below.

\[b_1 = r\cdot\frac{s_y}{s_x}\] \[b_0 = \bar{y} - b_1\cdot\bar{x}\]

VIT2005 %>% 
  summarize(r = cor(totalprice, area), 
            b1 = r*sd(totalprice)/sd(area), 
            b0 = mean(totalprice) - b1*mean(area))
          r       b1       b0
1 0.8092125 2704.751 40822.42
  • Print the least squares coefficients for mod1 using get_regression_table().
CT <- get_regression_table(mod1)
CT
# A tibble: 2 x 7
  term      estimate std_error statistic p_value lower_ci upper_ci
  <chr>        <dbl>     <dbl>     <dbl>   <dbl>    <dbl>    <dbl>
1 intercept   40822.    12170.      3.35   0.001   16835.   64810.
2 area         2705.      134.     20.2    0        2441.    2968.
CT[1, 2]
# A tibble: 1 x 1
  estimate
     <dbl>
1   40822.
CT[2, 2]
# A tibble: 1 x 1
  estimate
     <dbl>
1    2705.

1.2 Exercise

  • Create the linear model object mod2 by regressing totalprice on to elevatorF.

  • What totalprice do you predict for an apartment without an elevator and with an elevator (use the output from mod2 only).

  • Answer the previous question using group_by() and mean.

1.3 Exercise

  • Use the evals data set to create a parallel slopes model by regressing score on bty_avg and rank. Store the result in mod4.
mod4 <- lm(score ~ bty_avg + rank, data = evals)
mod4

Call:
lm(formula = score ~ bty_avg + rank, data = evals)

Coefficients:
     (Intercept)           bty_avg  ranktenure track       ranktenured  
         3.98155           0.06783          -0.16070          -0.12623

  • Write out the least squares regression lines for the three ranks.

  • Graph mod4 with ggplot (parallel slopes model).

2 Plotly and Directories

2.1 Creating a Directory

url <- "http://faculty.marshall.usc.edu/gareth-james/ISL/Advertising.csv"

if(!dir.exists("../Data/")){
dir.create("../Data/")
}

if(!file.exists("../Data/Advertising.csv")){
    download.file(url, destfile = "../Data/Advertising.csv")}
Ad <- read.csv("../Data/Advertising.csv")
head(Ad)
  X    TV radio newspaper sales
1 1 230.1  37.8      69.2  22.1
2 2  44.5  39.3      45.1  10.4
3 3  17.2  45.9      69.3   9.3
4 4 151.5  41.3      58.5  18.5
5 5 180.8  10.8      58.4  12.9
6 6   8.7  48.9      75.0   7.2
# Want to create a 3D scatterplot of sales ~ TV + Radio
library(plotly)
p <- plot_ly(data = Ad, z = ~sales, x = ~TV, y = ~radio, opacity = 0.6) %>%
  add_markers() 
p

Next we want to add the best fit plane to the 3D scatterplot. Consider the best fit plane sales_mod:

sales_mod <- lm(sales ~TV + radio, data = Ad)
summary(sales_mod)

Call:
lm(formula = sales ~ TV + radio, data = Ad)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.7977 -0.8752  0.2422  1.1708  2.8328 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  2.92110    0.29449   9.919   <2e-16 ***
TV           0.04575    0.00139  32.909   <2e-16 ***
radio        0.18799    0.00804  23.382   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.681 on 197 degrees of freedom
Multiple R-squared:  0.8972,    Adjusted R-squared:  0.8962 
F-statistic: 859.6 on 2 and 197 DF,  p-value: < 2.2e-16
x <- seq(0, 300, length = 70)
y <- seq(0, 50, length = 70)
plane <- outer(x, y, function(a, b){summary(sales_mod)$coef[1,1] + 
    summary(sales_mod)$coef[2,1]*a + summary(sales_mod)$coef[3,1]*b})
# draw the plane
p %>%
  add_surface(x = ~x, y = ~y, z = ~plane, showscale = FALSE)

Consider a model with interaction:

int_mod <- lm(sales ~ TV + radio + TV:radio, data = Ad)
summary(int_mod)

Call:
lm(formula = sales ~ TV + radio + TV:radio, data = Ad)

Residuals:
    Min      1Q  Median      3Q     Max 
-6.3366 -0.4028  0.1831  0.5948  1.5246 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 6.750e+00  2.479e-01  27.233   <2e-16 ***
TV          1.910e-02  1.504e-03  12.699   <2e-16 ***
radio       2.886e-02  8.905e-03   3.241   0.0014 ** 
TV:radio    1.086e-03  5.242e-05  20.727   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.9435 on 196 degrees of freedom
Multiple R-squared:  0.9678,    Adjusted R-squared:  0.9673 
F-statistic:  1963 on 3 and 196 DF,  p-value: < 2.2e-16
plane_int <- outer(x, y, function(a, b){summary(int_mod)$coef[1,1] + 
    summary(int_mod)$coef[2,1]*a + summary(int_mod)$coef[3,1]*b +
    summary(int_mod)$coef[4,1]*a*b})
p %>% 
  add_surface(x = ~x, y = ~y, z = ~plane_int, showscale = FALSE)

2.2 Exercise

Create a 3D representation of the best fit plane for the debt_model in Section 6.2.2 of Modern Dive.

library(ISLR)
head(Credit)
  ID  Income Limit Rating Cards Age Education Gender Student Married Ethnicity
1  1  14.891  3606    283     2  34        11   Male      No     Yes Caucasian
2  2 106.025  6645    483     3  82        15 Female     Yes     Yes     Asian
3  3 104.593  7075    514     4  71        11   Male      No      No     Asian
4  4 148.924  9504    681     3  36        11 Female      No      No     Asian
5  5  55.882  4897    357     2  68        16   Male      No     Yes Caucasian
6  6  80.180  8047    569     4  77        10   Male      No      No Caucasian
  Balance
1     333
2     903
3     580
4     964
5     331
6    1151
credit_ch6 <- Credit %>% 
  as_tibble %>% 
  select(ID, debt = Balance, credit_limit = Limit, income = Income, credit_rating = Rating, age = Age)
glimpse(credit_ch6)
Rows: 400
Columns: 6
$ ID            <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …
$ debt          <int> 333, 903, 580, 964, 331, 1151, 203, 872, 279, 1350, 140…
$ credit_limit  <int> 3606, 6645, 7075, 9504, 4897, 8047, 3388, 7114, 3300, 6…
$ income        <dbl> 14.891, 106.025, 104.593, 148.924, 55.882, 80.180, 20.9…
$ credit_rating <int> 283, 483, 514, 681, 357, 569, 259, 512, 266, 491, 589, …
$ age           <int> 34, 82, 71, 36, 68, 77, 37, 87, 66, 41, 30, 64, 57, 49,…
credit_ch6 %>% 
  select(debt, credit_limit, income) %>% 
  cor()
                  debt credit_limit    income
debt         1.0000000    0.8616973 0.4636565
credit_limit 0.8616973    1.0000000 0.7920883
income       0.4636565    0.7920883 1.0000000
p1 <- ggplot(credit_ch6, aes(x = credit_limit, y = debt)) + 
  geom_point(color = "red") +
  geom_smooth(method = "lm", se = FALSE) + 
  theme_bw() + 
  labs(x = "Credit limit (in $)",  y = "Credit card debt (in $)", 
       title = "Debt and credit limit")
p2 <- ggplot(data = credit_ch6, aes(x = income, y = debt)) + 
  geom_point(color = "red") + 
  geom_smooth(method = "lm", se = FALSE) +
  theme_bw() +
  labs(x = "Income (in $1000)", y = "Credit card debt (in $)",
       title = "Debt and income")
library(gridExtra)
grid.arrange(p1, p2, ncol = 2)

# Create a categorical variable with four levels for credit_limit
credit_ch6 <- credit_ch6 %>% 
  mutate(credit_limit_cat = cut(credit_limit, 
                                breaks = quantile(credit_limit, probs= c(0, .25, .5, .75, 1)), names = TRUE))

############
ggplot(data = credit_ch6, aes(x = income, y = debt, color = credit_limit_cat)) + 
  geom_point() + 
  geom_smooth(method = "lm", se = FALSE) +
  theme_bw() +
  labs(x = "Income (in $1000)", y = "Credit card debt (in $)",
       title = "Debt and income")

mod <- lm(debt ~ credit_limit + income, data = credit_ch6)
summary(mod)

Call:
lm(formula = debt ~ credit_limit + income, data = credit_ch6)

Residuals:
    Min      1Q  Median      3Q     Max 
-232.79 -115.45  -48.20   53.36  549.77 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -385.17926   19.46480  -19.79   <2e-16 ***
credit_limit    0.26432    0.00588   44.95   <2e-16 ***
income         -7.66332    0.38507  -19.90   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 165.5 on 397 degrees of freedom
Multiple R-squared:  0.8711,    Adjusted R-squared:  0.8705 
F-statistic:  1342 on 2 and 397 DF,  p-value: < 2.2e-16
p <- plot_ly(data = credit_ch6, z = ~debt, x = ~credit_limit, y = ~income, opacity = 0.8) %>%
  add_markers(marker = list(size = 8)) 
p
x <- seq(855, 13913, length = 70)
y <- seq(10, 187, length = 70)
plane <- outer(x, y, function(a, b){summary(mod)$coef[1,1] + 
    summary(mod)$coef[2,1]*a + summary(mod)$coef[3,1]*b})
# draw the plane
p %>%
  add_surface(x = ~x, y = ~y, z = ~plane, showscale = FALSE)

2.3 Interaction Model?

int_mod2 <- lm(debt ~ credit_limit + income + credit_limit:income, data = credit_ch6)
summary(int_mod2)

Call:
lm(formula = debt ~ credit_limit + income + credit_limit:income, 
    data = credit_ch6)

Residuals:
    Min      1Q  Median      3Q     Max 
-211.53 -110.14  -42.20   49.65  561.27 

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)         -3.071e+02  3.046e+01 -10.082  < 2e-16 ***
credit_limit         2.524e-01  6.839e-03  36.906  < 2e-16 ***
income              -9.785e+00  7.461e-01 -13.115  < 2e-16 ***
credit_limit:income  2.672e-04  8.082e-05   3.306  0.00103 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 163.4 on 396 degrees of freedom
Multiple R-squared:  0.8746,    Adjusted R-squared:  0.8736 
F-statistic: 920.4 on 3 and 396 DF,  p-value: < 2.2e-16
# int_plane2
int2_plane <- outer(x, y, function(a, b){summary(int_mod2)$coef[1,1] + 
    summary(int_mod2)$coef[2,1]*a + summary(int_mod2)$coef[3,1]*b + summary(int_mod2)$coef[4,1]*a*b})
# draw the plane
p %>%
  add_surface(x = ~x, y = ~y, z = ~int2_plane, showscale = FALSE)